General saddle cone beam CT apparatus and three-dimensional reconstruction method

ABSTRACT

A practical X-ray CT reconstruction method with a general saddle which overcomes restrictions in practical applications. A general saddle CT reconstruction method comprises: (a) placing an object relatively to a general saddle defined by the following conditions; (b) moving the X-ray source relatively to the object along the general saddle and detecting projections of the object on the detector plane; (c) filtering the projection along a predetermined filtering plane which is parallel to one of two major axes of the general saddle; and (d) performing three-dimensional reconstruction of the object through backprojection. The saddle curve is defined as such if the saddle curve is defined as such if a simple smooth curve which is single-connected, bounded and closed satisfies the following conditions against a family of continuous parallel planes in a three dimensional space: (1) for any plane in the family of parallel planes, there exist four different intersection points between the plane and the closed curve; (2) the four intersection points form a rectangle; and (3) rectangles obtained for any two different planes in the family of parallel planes are parallel to each other.

FIELD OF THE INVENTION

The present invention relates to an X-ray CT (Computed Tomography)method and apparatus, and more specifically to a three-dimensionalreconstruction method, apparatus and program from general saddle conebeam projections.

BACKGROUND OF THE INVENTION

X-ray CT (tomography) apparatus such as a circular cone-beam CTapparatus and a helical cone-beam CT apparatus as representativeexamples have been well known. Moreover, properties of standard saddlecurve (may be termed “saddle”, too) have been studied and reported(Non-patent document 1), and CT with a saddle has been proposed(Non-patent document-2). However, each of these methods and apparatushas certain disadvantages, respectively. Therefore, a novel andpractical method and apparatus are expected.

[Non-Patent Document 1]

G. L. Zeng, G. T. Gullberg, S. A. Foresti, “Eigen-analysis of cone-beamscanning geometries”, Proceedings of the 1995 Fully 3D Meeting(International Meeting on Fully Three-Dimensional Image Reconstructionin Radiography and Nuclear Medicine) p. 261-265

[Non-Patent Document 2]

J. D. Pack, F. Noo, H. Kudo, “Investigation of saddle trajectory forcardiac CT imaging in cone-beam geometry”, Proceedings of the 2003 Fully3D meeting. Phys. Med. Biol. 49, 2317-36, 2004

[Non-Patent Document 3]

Katsevich, A., “A general scheme for constructing inversion algorithmsfor cone beam CT”, Int. J. Math. Math. Sci. 21, 1305-21, 2003

SUMMARY OF THE DISCLOSURE

The disclosures of these references are incorporated herein by referencethereto.

There are the following problems for conventional CTs according to ouranalyses.

1. Circular cone beam CT: Since a circular trajectory does not satisfycone beam data completeness conditions (Tuy's condition), the quality ofthe reconstructed image deteriorates.

2. Helical cone beam CT: Since it requires overscan, it has problemssuch as increase in the measurement time and increase in X-ray exposureof objects. Moreover, serial imaging of data is difficult, because theX-ray source does not return to the original position after onerotation.

3. Properties of standard saddles have been studied by Zeng et al (1995)for the first time (Non-patent document 1). Thereafter, merits andproperties of Cardiac CT to diagnose heart disease with a saddle definedas a curve at the intersection of two surfaces s₁ and S₂ have beenstudied by Pack et al since 2003 (Non-patent document 2). However thestrict definition of the saddle trajectory by Pack et al excludes manyuseful curved trajectories. Moreover, the shift-variant CB-FBP imagereconstruction algorithm proposed by Pack et al is complex to implement.And it is difficult and requires much computation, because the filteringis not given by a one-dimensional convolution function.

Therefore, a CT scanner with a saddle trajectory has not been realizeduntil now, because it has been very difficult to obtain CT images with asaddle trajectory without simple and effective reconstructionalgorithms.

A reconstruction method desired from the general curved trajectories hasbeen applied to trajectories such as helix, circle and arc, circle andline (Non-patent document 3). However application to the saddletrajectories has not been discussed, because it is very difficult toverify the weighting equations in this method, and it is also verydifficult to implement the reconstruction algorithm with a simpleweighting. Moreover, it is unclear whether the method is applicable totruncation (whether the object is within the sight of camera or not).Until now, there has been no paper and report concerning thereconstruction with this method for the saddle trajectory.

According to an aspect of the present invention, it is an object of thepresent invention to provide a practical CT method and apparatus, andmore specifically a cone-beam CT method and apparatus with a generalsaddle curve (may be termed “saddle” too) which overcomes restrictionson saddles defined by Pack. According to another aspect of the presentinvention, it is an object to provide an accurate, effective and simplemethod and apparatus for three-dimensional reconstruction of objectsfrom projections (especially from projection data obtained with asaddle). Also it is an object of the present invention to provide aprogram therefor.

According to an aspect of the present invention, there is provided anovel X-ray CT method and apparatus. In this X-ray CT method andapparatus, the X-ray source (relative) trajectory is a general saddlewhich satisfies the cone-beam data completeness condition (Tuy'scondition). According to another aspect of the present invention, thereis provided an accurate, effective and simple three-dimensional imagereconstruction method for projection data obtained with this method orapparatus.

According to an aspect of the present invention, there is provided a newX-ray CT apparatus—a general saddle cone-beam CT (hereafter called“general saddle CT”) apparatus for CT imaging of short objects andpartial CT imaging of long objects. In the general saddle CT apparatus,the X-ray source (relative) trajectory in the X-ray source—detectionsystem is a general saddle which satisfies the cone-beam CT datacompleteness condition (Tuy's condition).

According to a first aspect of the present invention, there is provideda general saddle CT scan method (Mode 1) comprising:

(a) placing an object in association with a general saddle curve definedby the following conditions; and

(b) moving an X-ray source relatively to the object along the generalsaddle and capturing an X-ray projection of the object on a detectorplane;

wherein the saddle curve is a simple smooth curve which issingle-connected, bounded and closed, and satisfies the followingconditions against a family of continuous parallel planes in a threedimensional space:

(1) for any plane in the family of parallel planes, there exist fourdifferent intersection points between the plane and the closed curve;

(2) the four intersection points form a rectangle; and

(3) rectangles obtained for any two different planes in the family ofparallel planes are parallel to each other. (Mode 1)

According to a second aspect of the present invention, there is provideda general saddle CT scan and reconstruction methods (Mode 2), besidesthe above (a) and (b), further comprising:

(c) filtering the projection data along a predetermined filtering planewhich is parallel to one of two major axes of the general saddle curve;and

(d) performing three-dimensional reconstruction of the projectionthrough backprojection.

Hereafter, “the general saddle curve” or “the general saddle” means acurve defined later on.

According to a third aspect of the present invention, there is provideda general saddle CT scan apparatus (Mode 3). The general saddle CT scanapparatus comprises:

(a) a support unit which supports an object relatively to a generalsaddle curve defined by the conditions below;

(b) a driving unit for an X-ray source-object-detector system, whichprovides relative movement of an X-ray source and a detector plane alonga general saddle curve relatively to the object; and

(c) a capture unit which detects projection data of X-ray radiated fromthe X-ray source and penetrated through the object;

wherein the general saddle curve is a simple smooth curve which issingle-connected, bounded and closed, and satisfies the followingconditions against a family of continuous parallel planes in a threedimensional space:

(1) for any plane in the family of parallel planes, there exist fourdifferent intersection points between the plane and the closed curve;

(2) the four intersection points form a rectangle; and

(3) rectangles obtained for any two different planes in the family ofparallel planes are parallel to each other.

According to a fourth aspect of the present invention, there is provideda general saddle CT scan and reconstruction device (Mode 4), besides(a), (b) and (c) in the above third aspect, further comprising thefollowing (d) and (e):

(d) a filtering unit that filters the projection data along apredetermined filtering plane which is parallel to one of two major axesof the general saddle curve; and

(e) a backprojection and reconstruction unit that performsreconstruction of the object through backprojection.

According to a fifth aspect of the present invention, there is provideda computer program product for implementing and executing the abovemethods (Modes 1 and 2) by a computer. (Mode 5) The computer program maybe recorded on recording medium of any kind available and computer- ormachine-readable.

There are provided preferred modes of the present invention in thefollowing.

[Mode 6]

There is provided a general saddle CT scan and reconstruction methodaccording to the second aspect (Mode 2), wherein the filtering isperformed along an intersection line between the filtering plane and thedetector plane.

[Mode 7]

The filtering plane Π(λ,{right arrow over (θ)}) is chosen to satisfy thefollowing conditions in relation to a point {right arrow over (α)}(λ) ofthe X-ray source:

(1) the filtering plane passes through the point {right arrow over(α)}(λ);

(2) the filtering plane is parallel to a vector {right arrow over(θ)}=(θ_(x),θ_(y),θ_(z))^(T) extending from the point {right arrow over(α)}(λ) to a point on the detector plane; and

(3) the filtering plane is parallel to a vector {right arrow over(ε)}(λ,θ), where the vector {right arrow over (e)}(λ,{right arrow over(θ)}) is chosen to satisfy the following condition:${\overset{\rightarrow}{e}\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} = \left\{ \begin{matrix}{{{\overset{\rightarrow}{e}}_{x}\quad{if}\quad\theta_{z}} \geq 0} \\{{{\overset{\rightarrow}{e}}_{y}\quad{if}\quad\theta_{z}} < 0.}\end{matrix} \right.$[Mode 8]

The filtering is performed by the following equation:${{g_{F}\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} = \left. {\int_{- \pi}^{\pi}{{\mathbb{d}\gamma}\quad\frac{1}{\sin\quad\gamma}\frac{\partial}{\partial\overset{\_}{\lambda}}{g\left( {\overset{\_}{\lambda},{\xi\left( {\lambda,\theta,\gamma} \right)}} \right)}}} \right|_{\overset{\_}{\lambda} = \lambda}},{\theta \in S^{2}}$where${{\overset{\rightarrow}{\xi}\left( {\lambda,\overset{\rightarrow}{\theta},\gamma} \right)} = {{\cos\quad\gamma\quad\overset{\rightarrow}{\theta}} + {\sin\quad\gamma\quad\frac{{\overset{\rightarrow}{e}\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} - {\left( {{\overset{\rightarrow}{e}\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} \cdot \overset{\rightarrow}{\theta}} \right)\overset{\rightarrow}{\theta}}}{{{\overset{\rightarrow}{e}\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} - {\left( {{\overset{\rightarrow}{e}\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} \cdot \overset{\rightarrow}{\theta}} \right)\overset{\rightarrow}{\theta}}}}}}},$and g(λ,{right arrow over (ξ)}) denotes the projection data.[Mode 9]

The backprojection is performed by the following equation:${{f\left( \overset{\rightarrow}{x} \right)} = {{- \frac{1}{4\pi^{2}}}{\int_{0}^{2\pi}{\frac{\mathbb{d}\lambda}{{\overset{\rightarrow}{x} - {\overset{\rightarrow}{a}(\lambda)}}}{g_{F}\left( {\lambda,\frac{\overset{\rightarrow}{x} - {\overset{\rightarrow}{a}(\lambda)}}{{\overset{\rightarrow}{x} - {\overset{\rightarrow}{a}(\lambda)}}}} \right)}}}}},$where f({right arrow over (x)}) denotes CT value at each point of theobject.[Mode 10]

A well-designed general saddle curve defined in the following isemployed as the general saddle curve, in case where the object is long,and the reconstruction region FOV is sandwiched between two (horizontal)planes crossing the object. The general saddle curve is termed“well-designed” subject to the following conditions:

(1) left (or right) part of a given general saddle curve above a first(horizontal) plane is located on the left (or right) side of thereconstruction region FOV;

(2) front (or back) part of a given general saddle curve below a second(horizontal) plane is located on the front (or back) side of a front (orback) plane of the reconstruction region FOV.

[Model 11]

A three-dimensional reconstruction of the object is performed byrepeating the steps (c) and (d) for each value of the parameter λ of thegeneral saddle curve.

[Mode 12]

The filtration comprises differentiating the projection data withrespect to the parameter λ of the general saddle curve, weighting afterthe differentiation, interpolation before filtering, and 1D Hilberttransform of the interpolated data.

[Mode 13]

Filtration and coordinate transformation of the filtered data areperformed after the 1D Hilbert transform.

[Mode 14]

The backprojection is performed on the filtered data after the 1DHilbert transform.

[Mode 15]

The filtration is performed by defining two weighted data based on theprojection data, and the reconstruction is performed using these datawithout differentiation operation with respect to the parameter λ of thegeneral saddle curve.

[Mode 16]

The well-designed general saddle can be implemented by restricting valueabout one axis among two rotation axes in C-arm geometry or ringgeometry with respect to a value of the other axis.

[Mode 17]

The general saddle curve is established by rotating the object (sample)about a rotation axis thereof, fixing either one of the X-ray source andthe detector plane, and controlling their relative positionalrelationship.

[Mode 18]

The general saddle curve is established by fixing the X-ray source andthe detector plane, rotating the object (sample) about its rotation axisand moving the object forward and reverseward (upward and downward)along the direction of its rotation axis in accordance with apredetermined equation.

The meritorious effects of the present invention are summarized asfollows.

The general saddle CT apparatus according to the present inventionprovides a more (theoretically) accurate reconstruction of objects thanthe conventional circular cone-beam CT apparatus which does not satisfythe cone-beam CT data completeness condition. Since the general saddleCT apparatus eliminates the problem of overscan for CT imaging of ashort object and partial CT imaging of a long object, it requiresshorter measurement time and less X-ray exposure of objects than thehelical cone-beam CT apparatus. Moreover, according to the presentinvention, an accurate, effective and simple three-dimensional imagereconstruction method for the projection data obtained in the generalsaddle CT apparatus is provided. The reconstruction method iscategorized to the FBP (filtered backprojection)-type reconstructionmethod in the same way as the FDK-type method which can be easilyimplemented. Therefore, the present invention provides the firsteasy-to-implement FBP-type reconstruction method, and solves problemsconcerning CT imaging with the general saddle X-ray source trajectory.

To solve problems in CT imaging of short objects and partial CT imagingof long objects, general saddles or saddle trajectories provide an X-raysource scan method for the CT apparatus according to the presentinvention.

[Definition of Terms]

The present invention employs novel concepts like “a general saddlecurve (or trajectory)”, “a reconstruction region FOV (Field of View)”,“its front, back, left and right planes” and “a well-defined generalsaddle curve (or “saddle”). Therefore, these new concepts are defined inthe following.

(1) The definition of a “general saddle curve (trajectory)”

The general saddle curve is a simple smooth curve which issingle-connected, bounded and closed satisfies the following conditionsagainst a family of continuous parallel planes in the three dimensionalspace:

(i) for any plane in the family of parallel planes, there exist fourdifferent intersection points between the plane and the closed curve;

(ii) these four intersection points form a rectangle; and

(iii) rectangles obtained for any two different planes in the family ofparallel planes are parallel to each other.

(2) The definition of a reconstruction region FOV (field of view) andits front, back, left, and right planes

A reconstruction region FOV is defined as a part sandwiched (enclosed)between two (horizontal) parallel planes A and B when the object is long(e.g., shaped like a long column or cylinder).

A front plane of the reconstruction region FOV is the closest plane tothe object in a set of vertical (orthogonal to the y-axis) planes infront of the object (in the negative direction of the y-axis).

A back plane of the reconstruction region FOV is the closest plane tothe object in a set of vertical (orthogonal to the y-axis) planes on theback side of the object (in the positive direction of the y-axis).

A left plane of the reconstruction region FOV is the closest plane tothe object in a set of vertical (orthogonal to the x-axis) planes on theleft side of the object (in the negative direction of the x-axis).

A right plane of the reconstruction region FOV is the closest plane tothe object in a set of vertical (orthogonal to the x-axis) planes on theright side of the object (in the positive direction of the x-axis).

Note that the terms “front, back, left and right” here are definedaccording to FIG. 1 with respect to the Cartesian coordinate system(x,y,z).

(3) Definition of a well-designed general saddle curve (trajectory)

A set of well-designed general saddle curves is defined according to thereconstruction region FOV and it is subset of a set of general saddlecurves. In case where the object is long, and the reconstruction regionFOV is sandwiched between two (horizontal) planes crossing the object,the general saddle curve is termed “well-designed” if it satisfies thefollowing conditions:

(1) One part, i.e., left (or right) part of a given general saddle curveabove the first (horizontal) plane (A) is located on the left (or right)side of the left (or right) plane of the reconstruction region FOV.

(2) One part, i.e., front (or back) part of a given general saddle curvebelow the second (horizontal) plane (B) is located on the front (orback) side of the front (or back) plane of the reconstruction regionFOV.

BRIEF DESCRIPTIONS OF THE DRAWINGS

FIG. 1 schematically shows a general saddle trajectory and areconstruction region FOV.

FIG. 2 schematically shows the definition of a general saddle trajectoryand its relation to the filtering plane and the detector plane.

FIG. 3 shows the relation between the coordinate for the general saddletrajectory and the coordinate for the projection data on the detectorplane.

FIG. 4 shows the relation between the coordinate for the general saddleand the coordinate for the projection data on the detector plane for aC-arm geometry.

FIG. 5 shows an implementation example of a general saddle trajectorywith a fixed X-ray source and a fixed detector.

FIG. 6 shows an implementation example of a general saddle trajectorywith a fixed X-ray source and a movable detector.

FIG. 7 shows C-arm geometry which can be rotated about two major axes(λ,α).

FIG. 8 shows geometry for reconstructing a thin (or flat) object with ageneral saddle curve.

FIG. 9 shows a block diagram of an apparatus according to the presentinvention.

FIG. 10 shows results from Example 6. It shows cross sections of 3Dreconstructed images by the VD algorithm and the VI algorithm using aShepp-Logan phantom. The parameters for a standard saddle curve inExample 1 are R=12 cm, D=12 cm, h=4 cm, and the detector plane is at 7.2cm.

FIG. 11 shows results from Example 7. It shows cross sections of 3Dreconstructed images by the VD algorithm and the VI algorithm using adisc phantom. The parameters for general saddle curve in Example 1 areR=12 cm, D=12 cm, h=4 cm, and the detector plane is at 7.2 cm.

FIG. 12 shows results from Example 8. It shows cross sections of 3Dreconstructed images by the VD algorithm and the VI algorithm usingShepp-Logan phantom. The parameters for C-arm geometry in Example 3 areR=7 cm, D=14 cm, k=0.3 cm, and the detector is at 7.2 cm.

PREFERRED MODES OF THE INVENTION

1. The present invention employs a general saddle scan method. This is anovel X-ray CT method. A set of general saddle curves proposed in thepresent invention provides a more comprehensive set of curves whichincludes conventional sets of saddle curves and has the merit ofapplicability to Implementation of many kinds of hardware.

Note that the present invention includes a case where a combination ofmovement or rotation of the X-ray source, detector and the object hasthe same geometry as that of the general saddle trajectory in the imagereconstruction space.

2. The present invention makes it possible to use a novel filteredbackprojection reconstruction method (algorithm) in a general saddle CTapparatus. The reconstruction method has the merit of accuracy, highperformance and easiness in implementation. The steps of thereconstruction method can be summarized in the following three steps.

Step 1: Differentiation of measured projection data along the generalsaddle curve.

Step 2: Filtering the differentiated projection data along a direction(a predetermined filtering plane) specified in the present invention.

Step 3: Backprojection of the filtered projection data.

The main feature of the reconstruction method (or algorithm) accordingto the present invention is that the filtering direction is specified ona plane which is parallel to one of two major axes (in the orthogonalcoordinate system) of the general saddle curve.

3. According to the present invention, detailed steps for reconstructionalgorithms are presented in accordance with each geometry for somevarious special examples of general saddles. Moreover, some modifiedalgorithms which do not include differentiation along general saddlecurves are also presented. These modifications are numerically morerobust and reduce artifacts in the reconstructed images.

4. According to the present invention, hardware implementations of somespecial general saddle curves and their applicability are considered.

1. The Definition of a General Saddle Trajectory

[Definition]

Although the general saddle curve (trajectory) has been definedconceptually in the above “Definition of Terms” (1), it can be definedmore exactly in mathematical terms as follows.

The “general saddle curve” is defined as such if a simple smooth curve Cwhich is single-connected, bounded and closed satisfies the followingconditions against a family of continuous parallel planes Π({tilde over(z)}){tilde over (z)}_(min)<{tilde over (z)}<{tilde over (z)}_(max):

(1) for any {tilde over (z)}_(min)<{tilde over (z)}<{tilde over(z)}_(max), there exist four different intersection points between theplane Π({tilde over (z)}) and a curve C, where the four intersectionpoints are denoted as A_(k)({tilde over (z)}),(k=1,2,3,4).

(2) A_(k)({tilde over (z)}),(k=1,2,3,4) are continuous functions ofparameter {tilde over (z)}, and A_(k)({tilde over (z)}),(k=1,2,3,4) forma rectangle for any parameter {tilde over (z)}.

(3) rectangles obtained for any two different value of the parameter{tilde over (z)} are parallel. This means that the direction ofA₁({tilde over (z)}₁)A₂({tilde over (z)}₁) and that of A₁({tilde over(z)}₂)A₂({tilde over (z)}₂) for any two different {tilde over (z)}₁ and{tilde over (z)}₂ are identical, and the same holds for A₂({tilde over(z)}₁)A₃({tilde over (z)}₁) and A₂({tilde over (z)}₂)A₃({tilde over(z)}₁) A₄({tilde over (z)}₁) and A₃({tilde over (z)}₂)A₄({tilde over(z)}₂), A₄({tilde over (z)}₁)A₁({tilde over (z)}₁) and A₄({tilde over(z)}₂)A₁({tilde over (z)}₂), respectively.

In the present invention, the directions of A₁({tilde over(z)})A₂({tilde over (z)}) and A₂({tilde over (z)})A₃({tilde over (z)})are termed “two major directions” of the general saddle. These majordirections are independent of the value {tilde over (z)}.

2. Coordinate system and parameterization of the general saddle

2.1 A point O is the center of a rectangle A₁({tilde over(z)}₀)A₂({tilde over (z)}₀)A₃({tilde over (z)}₀)A₄({tilde over (z)}₀),where${\overset{\sim}{z}}_{0} = {\frac{1}{2}{\left( {{\overset{\sim}{z}}_{\min} + {\overset{\sim}{z}}_{\max}} \right).}}$

Here, a three dimensional space coordinate system with its origin at thepoint O is defined.

The unit vector {right arrow over (e)}_(x) along the x-axis is given by${{\overset{\rightarrow}{e}}_{x} = \frac{{A_{2}\left( {\overset{\sim}{z}}_{0} \right)} - {A_{1}\left( {\overset{\sim}{z}}_{0} \right)}}{{{A_{2}\left( {\overset{\sim}{z}}_{0} \right)} - {A_{1}\left( {\overset{\sim}{z}}_{0} \right)}}}},$the unit vector {right arrow over (e)}_(y) along the y-axis is given by${{\overset{\rightarrow}{e}}_{y} = \frac{{A_{3}\left( {\overset{\sim}{z}}_{0} \right)} - {A_{2}\left( {\overset{\sim}{z}}_{0} \right)}}{{{A_{3}\left( {\overset{\sim}{z}}_{0} \right)} - {A_{2}\left( {\overset{\sim}{z}}_{0} \right)}}}},$and the unit vector {right arrow over (e)}_(z) along the z-axis is givenby {right arrow over (e)}_(z)={right arrow over (e)}_(x)×{right arrowover (e)}_(y).2.2 In the coordinate system (x,y,z), the family of parallel planesΠ({tilde over (z)}){tilde over (z)}_(min)<{tilde over (z)}<{tilde over(z)} can be parameterized by a parameter {tilde over (z)}. In thepresent invention, Π({tilde over (z)}),{tilde over (z)}_(min)<{tildeover (z)}<{tilde over (z)}_(max) denotes the family of parallel planes.2.3 Using the Cartesian coordinate system (x,y,z), the general saddletrajectory C is parameterized as{right arrow over (α)}(λ)=(α_(x)(λ),α_(y)(λ),α_(z)(α)),0≦λ≦27π.2.4 For a given value for parameter z, the parameterized general saddletrajectory C and Π(z) have four intersection points parameterized with λas given by{right arrow over (α)}(λ₁ ⁽⁼⁰⁾)=A ₁, {right arrow over (α)}(λ₂ ⁽⁼⁾ =A ₂,{right arrow over (α)}(λ₃ ⁽⁼⁾)=A ₃, {right arrow over (α)}(λ₄ ⁽⁼⁾)=A ₄.

Moreover the following conditions are assumed: $\left\{ \begin{matrix}{{a_{z}(\lambda)} \leq z} & {if} & {\lambda \in {\left( {\lambda_{1}^{(z)},\lambda_{2}^{(z)}} \right)\bigcup{\left( {\lambda_{3}^{(z)},\lambda_{4}^{(z)}} \right).}}} \\{{a_{z}(\lambda)} \geq z} & {if} & {\lambda \in {\left( {\lambda_{2}^{(z)},\lambda_{3}^{(z)}} \right)\bigcup{\left( {\lambda_{4}^{(z)},\lambda_{1}^{(z)}} \right).}}}\end{matrix}\quad \right.$

In the present invention, x- and y-axes are called major axes of thegeneral saddle.

2.5 In the present invention, we define a set of points asΩ(C)={{right arrow over (x)}ε A ₁A₂A₃A₄ (z)|z _(min) <z<z _(max)}.where A₁A₂A₃A₄ (z) is a set of inner and boundary points of therectangle A₁A₂A₃A₄ for fixed value of z. We also define Ω⁰(C)={{rightarrow over (x)}εΩ(C){right arrow over (x)} is an inner point of Ω(C)}.

The set of points Ω⁰(C) satisfies the cone-beam data completenesscondition (Tuy's condition). We assume that the reconstruction regionfiled of view FOV defined asFOV={(x,y,z)|x ² +y ² ≦B ² ,z ^(b) _(min)≦z≦z^(b)}is a subset of Ω⁰(C).<The Definition of a Well-Designed General Saddle Trajectory>

A general saddle is called well-designed relating to the FOV if itsatisfiesα_(y)(λ)<−B if λε(λ₁ ⁼ ^(min) ^(b) ⁾,λ₂ ⁽⁼ ^(min) ^(b) ⁾),α_(y)(λ)>B if λε(λ₃ ⁽⁼ ^(min) ^(b) ⁾,λ₄ ⁽⁼ ^(min) ^(b) ⁾,α_(x)(λ)>B if λε(λ₂ ⁽⁼ ^(max) ^(b) ⁾,λ₃ ⁽⁼ ^(min) ^(b) ⁾,α_(x)(λ)<−B if λε(λ₄ ⁽⁼ ^(max) ^(b) ⁾, λ₁ ⁽⁼ ^(max) ^(b) ⁾).3. The set of general saddles includes ordinary saddles3.1 A saddle trajectory is defined as the curve established at theintersection of two surfaces (S₁, S₂):S ₁={(x,y,z):z=s ₁(x)} andS ₂={(x,y,z):z=s ₂(y)}where the two functions S₁(x), and S₂(y) satisfy the followingconditionss ₁″(x)>0, s ₁′(0)=0,s₁(0)<0 for any x,s ₂″(y)<0,s ₂′(0)=0, s ₂(0)=−s ₁(0) for any y.3.2 Since a family of planes Π(z),s₁(0)<z<s₂(0) parallel to the(x,y)-plane satisfies the conditions of the general saddle, the saddletrajectory defined above is a special example of the general saddles.All ordinary saddles are well-designed general saddles.3.3 An example of a general but not ordinary saddleCurve on the Sphere

A curve defined by (R cos(α(λ))cos λ,R cos(α(λ))sin λ,R sin(α(λ))^(T)where${\alpha(\lambda)} = {\tan^{- 1}\frac{k\quad\cos\quad 2\lambda}{\sqrt{1 - k^{2}}}}$is an ordinary saddle when 0<k<0.5. However, the curve is not anordinary saddle but a general saddle when 0.5≦k<1. Moreover, this curveis also a well-designed general saddle when 0.5≦k<1.4. The definition of cone beam projection data${{g\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} = {\int_{0}^{\infty}{{\mathbb{d}t}\quad{f\left( {{\overset{\rightarrow}{a}(\lambda)} + {t\quad\overset{\rightarrow}{\theta}}} \right)}}}},{\overset{\rightarrow}{\theta} \in S^{2}},$where S² is the unit sphere.

5. The definition of a filter/correction backprojection operator5.1 Differentiating${g^{\prime}\left( {\lambda,\overset{\rightarrow}{\theta}} \right)} = {\frac{\partial}{\partial\lambda}{{g\left( {\lambda,\overset{\rightarrow}{\theta}} \right)}.}}$5.2 Filtering

The filtering plane is composed of following three elements, i.e., theposition of the X-ray source {right arrow over (α)}(λ), the point to bereconstructed {right arrow over (x)} and the direction {right arrow over(e)}. The filtering plane passes through a point {right arrow over(α)}(λ) and a point {right arrow over (x)}, and is parallel to a vector{right arrow over (e)}. We consider following two orthogonal vectors inthe filtering plane,${{\overset{\rightarrow}{\alpha}\left( {\lambda,\overset{\rightarrow}{x}} \right)} = \frac{\overset{\rightarrow}{x} - {\overset{\rightarrow}{a}(\lambda)}}{{\overset{\rightarrow}{x} - {\overset{\rightarrow}{a}(\lambda)}}}},{{\overset{\rightarrow}{\beta}\left( {\lambda,\overset{\rightarrow}{x},\overset{\rightarrow}{e}} \right)} = {\frac{\overset{\rightarrow}{e} - {\left( {\overset{\rightarrow}{e} \cdot {\overset{\rightarrow}{\alpha}\left( {\lambda,\overset{\rightarrow}{x}} \right)}} \right){\overset{\rightarrow}{\alpha}\left( {\lambda,\overset{\rightarrow}{x}} \right)}}}{{\overset{\rightarrow}{e} - {\left( {\overset{\rightarrow}{e} \cdot {\overset{\rightarrow}{\alpha}\left( {\lambda,\overset{\rightarrow}{x}} \right)}} \right){\overset{\rightarrow}{\alpha}\left( {\lambda,\overset{\rightarrow}{x}} \right)}}}}.}}$

Any direction in the filtering plane can be parameterized by an angle γfrom the vector {right arrow over (α)} and given by{right arrow over (θ)}(λ,{right arrow over (x)},{right arrow over(e)},γ)=cos γ·{right arrow over (α)}(λ,{right arrow over (x)})+sinγ·{right arrow over (β)}(λ,{right arrow over (x)},{right arrow over(e)}).

The filtering operator is defined by${g_{F}\left( {\lambda,\overset{\rightarrow}{x},\overset{\rightarrow}{e}} \right)} = {\int_{- \pi}^{\pi}{{\mathbb{d}\gamma}\quad\frac{1}{\sin\quad\gamma}{g^{\prime}\left( {\lambda,{\overset{\rightarrow}{\theta}\left( {\lambda,\overset{\rightarrow}{x},\overset{\rightarrow}{e},\gamma} \right)}} \right)}}}$

This operator is essentially a one-dimensional (1D) Hilbert transform onthe intersection line of the filtering plane and the detector plane.5.3 The backprojection is applied to the filtered result afterweighting.${K\left( {\overset{\rightarrow}{x},\overset{\rightarrow}{e},\lambda^{-},\lambda^{+}} \right)} = {{- \frac{1}{2\pi^{2}}}{\int_{\lambda^{-}}^{\lambda^{+}}{{\mathbb{d}\lambda}\quad\frac{1}{{\overset{\rightarrow}{x} - {\overset{\rightarrow}{a}(\lambda)}}}{g_{F}\left( {\lambda,\overset{\rightarrow}{x},\overset{\rightarrow}{e}} \right)}}}}$

If λ⁻>λ⁻, we regard (approximate) λ⁺ as λ⁺+2π in the above integration.

6. Reconstruction formula

For any point {right arrow over (x)}=(x,y,z₀)^(T) within the FOV, theplane z=z₀ and the general saddle trajectory intersect at four points{right arrow over (α)}(λ₁ ⁽⁼ ⁰ )⁾, {right arrow over (α)}(λ₂ ⁽⁼ ⁰⁾),{right arrow over (α)}(λ₃ ⁽⁼ ⁰ ⁾),{right arrow over (α)}(λ₄ ⁽⁼ ⁰ ⁾).Therefore the reconstruction formula at a point {right arrow over(x)}=(x,y,z₀)^(T) is given by${f\left( \overset{\rightarrow}{x} \right)} = {\frac{1}{2}{\begin{Bmatrix}{{K\left( {\overset{\rightarrow}{x},{\overset{\rightarrow}{e}}_{x},\lambda_{1}^{(z_{0})},\lambda_{2}^{(z_{0})}} \right)} + {K\left( {\overset{\rightarrow}{x},{\overset{\rightarrow}{e}}_{y},\lambda_{2}^{(z_{0})},\lambda_{3}^{(z_{0})}} \right)} +} \\{{K\left( {\overset{\rightarrow}{x},{- {\overset{\rightarrow}{e}}_{x}},\lambda_{3}^{(z_{0})},\lambda_{4}^{(z_{0})}} \right)} + {K\left( {\overset{\rightarrow}{x},{- {\overset{\rightarrow}{e}}_{y}},\lambda_{4}^{(z_{0})},\lambda_{1}^{(z_{0})}} \right)}}\end{Bmatrix}.}}$7. FBP-type implementation of the reconstruction method and itsapplicability to the axial truncation data

Superficially, the filtering plane is related to the point {right arrowover (x)}=(x,y,z₀)^(T) to be reconstructed in the above reconstructionformula and the point {right arrow over (x)}=(x,y,z₀)^(T) is needed inthe filtering process. Therefore, it seems as if numericalimplementation necessary for the reconstruction be very complex andtotally different from the computation scheme of the FBP type.

However, in the well-designed general saddle, the filtering surface inthe above reconstruction formula can be parameterized in one dimensionas follows, in case where the projection angle λ is fixed.

Denote the intersection point between the filtering plane and the z-axisas (0,0,z)^(T). A plane parallel to the x-axis is selected as afiltering plane if z>α₌(λ), and a plane parallel to the y-axis isselected as a filtering plane if z≦α₌(λ). This family of filteringplanes is denoted by π(z).

From the definition of the coordinate for the general saddle trajectory,all filtering planes for the reconstruction formula are contained in thefamily π(z). Therefore the filtering can be performed prior tobackprojection in the present invention. This filtering process isindependent of the position of the point {right arrow over (x)}=(x,y,z₀)to be reconstructed and shares the same basic properties with theconventional FBP-type algorithms.

Also note that any plane in the family π(z) is not parallel to thez-axis for a well-designed general saddle trajectory, so the algorithmof the present invention is applicable to axial truncation data.

8. Special Example 1 (detailed reconstruction steps)

8.1 Definition

We consider a class of well-designed general saddles parameterized as(R(λ)cos λ, R(λ)sin λ, α₌(λ))^(T),where λ is the polar angle in the (x,y)-plane and the polar radius R(λ)satisfies R(λ)>0.8.2 Detector Geometry

In the reconstruction space, the coordinate for the detector is given by$\left\{ \begin{matrix}{{{\overset{\rightarrow}{e}}_{u}(\lambda)} = \left( {{{- \sin}\quad\lambda},{\cos\quad\lambda},0} \right)^{T}} \\{{{\overset{\rightarrow}{e}}_{v}(\lambda)} = \left( {{{- \cos}\quad\lambda},{{- \sin}\quad\lambda},0} \right)^{T}} \\{{{\overset{\rightarrow}{e}}_{w}(\lambda)} = \left( {0,0,1} \right)^{T}}\end{matrix}\quad \right.$

The detector plane is arranged to be orthogonal to the unit vector{right arrow over (α)}_(v)(λ). The origin of the detector plane isplaced at the projection of the point {right arrow over (α)}(λ) onto thedetector plane. Cartesian coordinates (u,w) specify a pixel in thedetector plane. The u-axis is parallel to the vector {right arrow over(e)}_(u), and the w-axis is parallel to the vector {right arrow over(e)}_(w).

The distance between the detector plane and the X-ray source is denotedby D(λ)(>0), and the projection data on the detector plane is denoted byg_(f)(λ,u,w).

8.3 Reconstruction Steps

STEP 1: Filtering.

Each projection data g_(f)(λ,u,w) is modified/transformed into g_(f)^(F)(λ,u,w) according to the following filtering steps:FF1: derivative at constant direction. Compute${g_{1}\left( {\lambda,u,w} \right)} = {\left( {\frac{\partial}{\partial\lambda} + {\frac{{{D^{\prime}(\lambda)}u} + u^{2} + {D^{2}(\lambda)}}{D(\lambda)}\frac{\partial}{\partial u}} + {\frac{{{D^{\prime}(\lambda)}w} + {uw}}{D(\lambda)}\frac{\partial}{\partial w}}} \right){{g_{f}\left( {\lambda,u,w} \right)}.}}$FF2: length-correction weighting. Compute${g_{2}\left( {\lambda,u,w} \right)} = {\frac{D(\lambda)}{\sqrt{{D^{2}(\lambda)} + u^{2} + w^{2}}}{{g_{1}\left( {\lambda,u,w} \right)}.}}$FF3: interpolation before filtering. Computeg ₃(λ,u,z)=g ₂(λ,u,w(u,z))where z is a parameter for the filtering plane π(z), and w(u,z) is theintersection line between the filtering plane π(z) and the detectorplane.FF4: 1D Hilbert transformComputeg ₄(λ,u,z)=∫_(−∞) ^(∞) du′h _(H)(u−u′)g ₃(λ,u′,z),where h_(H) is the impulse response of the Hilbert transform.FF5: Transform after filteringComputeg _(f) ^(F)(λ,u,w)=g ₄(λ,u,z(u,w)),where z(u,w) denotes the parameter for the filtering plane which passesthrough the point (u,w) on the detector plane.STEP 2: Backprojection

The filtered projection g_(f) ^(F)(λ,u,w) is backprojected to formulatef at each point {right arrow over (x)}=(x,y,z) of the FOV according tothe formula${{f\left( \overset{\rightarrow}{x} \right)} = {\frac{1}{4\pi}{\int_{0}^{2\pi}{{\mathbb{d}\lambda}\quad\frac{1}{\overset{\rightarrow}{v}\left( {\overset{\rightarrow}{x},\lambda} \right)}{g_{f}^{F}\left( {\lambda,{\overset{\_}{u}\left( {\overset{\rightarrow}{x},\lambda} \right)},{\overset{\_}{w}\left( {\overset{\rightarrow}{x},\lambda} \right)}} \right)}}}}},$where ({right arrow over (u)}({right arrow over (x)},λ),{right arrowover (w)}({right arrow over (x)},λ)) denote the coordinates of theintersection point of the detector plane and the line connecting thepoints {right arrow over (x)} and {right arrow over (α)}(λ), and{right arrow over (v)}({right arrow over (x)},λ)=R(λ)−x cos λ−y sin λ.

Note that by eliminating the procedure FF5 of STEP 1 in the abovereconstruction steps and incorporating the corresponding procedure inthe backprojection of STEP 2, resolution of the reconstructed image canbe increased.

8.4 The above reconstruction step in 8.3 needs differentiation withrespect to the parameter λ for the general saddle curve. Thedifferentiation processing increases sampling and discretization errors,which are counted as drawbacks.

There are some modified formulae which eliminate the differentiationoperation with respect to λ from the reconstruction formula at 8.3.Here, an example thereof is shown.

First, we define two weighted data by${{{\overset{\sim}{g}}_{f}\left( {\lambda,u,w} \right)} = {\frac{{D(\lambda)} + {{{uR}^{\prime}(\lambda)}/{R(\lambda)}}}{\sqrt{{D^{2}(\lambda)} + u^{2} + w^{2}}}{g_{f}\left( {\lambda,u,w} \right)}}},{and}$${{\overset{\Cap}{g}}_{f}\left( {\lambda,u,w} \right)} = {\frac{D(\lambda)}{\sqrt{{D^{2}(\lambda)} + u^{2} + w^{2}}}{{g_{f}\left( {\lambda,u,w} \right)}.}}$

Then, the reconstruction formula is given as follows:${{f\left( \overset{\rightarrow}{x} \right)} = \left. {{- \frac{1}{4\pi^{2}}}{\int_{0}^{2\pi}{{\mathbb{d}\lambda}\frac{{D(\lambda)}{R(\lambda)}}{{\overset{\_}{v}}^{2}}{\int{\frac{\mathbb{d}u}{\left( {\overset{\_}{u} - u} \right)^{2}}{{\overset{\sim}{g}}_{f}\left( {\lambda,u,w} \right)}}}}}} \middle| {}_{w = {\overset{\_}{w} + {\kappa{({u - \overset{\_}{u}})}}}}{{- \frac{1}{4\pi^{2}}}{\int_{0}^{2\pi}{{\mathbb{d}\lambda}\frac{{{D(\lambda)}\left( {{\kappa\quad{R(\lambda)}} - {a_{z}^{\prime}(\lambda)}} \right)} - {{R^{\prime}(\lambda)}\left( {\overset{\_}{w} - {\kappa\quad\overset{\_}{u}}} \right)}}{{\overset{\_}{v}}^{2}}{\int{\frac{\mathbb{d}u}{\overset{\_}{u} - u}\left( {\frac{\partial}{\partial w}{{\overset{\Cap}{g}}_{f}\left( {\lambda,u,w} \right)}} \right)}}}}} \middle| {}_{w = {\overset{\_}{w} + {\kappa{({u - \overset{\_}{u}})}}}}{{- \frac{1}{4\pi^{2}}}{\int_{0}^{2\pi}{\frac{{{D(\lambda)}\left( {{\kappa\quad{R(\lambda)}} - {a_{z}^{\prime}(\lambda)}} \right)} - {{R^{\prime}(\lambda)}\left( {\overset{\_}{w} - {\kappa\quad\overset{\_}{u}}} \right)}}{{\overset{\_}{v}}^{2}}{\chi\left( {\lambda,\overset{\_}{u}} \right)}{\int{\mathbb{d}{u\left( {\frac{\partial}{\partial w}{{\overset{\Cap}{g}}_{f}\left( {\lambda,u,w} \right)}} \right)}}}}}} \right|_{w = {\overset{\_}{w} + {\kappa{({u - \overset{\_}{u}})}}}}},$where κ is the gradient of the intersection line between the filteringplane and the detector plane, and χ is defined by${\chi\left( {\lambda,\overset{\_}{u}} \right)} = \left\{ \begin{matrix}\frac{\cos\quad\lambda}{{{D(\lambda)}\quad\sin\quad\lambda} - {\overset{\_}{u}\cos\quad\lambda}} & {{{if}\quad\overset{\_}{w}} > 0} \\\frac{{- \sin}\quad\lambda}{{{D(\lambda)}\quad\cos\quad\lambda} + {\overset{\_}{u}\sin\quad\lambda}} & {{{if}\quad\overset{\_}{w}} < 0.}\end{matrix} \right.$8.5 Another example of reconstruction formula which eliminates thedifferentiation operation with respect to the parameter λ of thewell-designed general saddle curve from the reconstruction formula isgiven as follows:${{f\left( \overset{->}{x} \right)} = \left. {\frac{1}{4\quad\pi^{2}}{\int_{0}^{2\quad\pi}{{\mathbb{d}\lambda}\quad\frac{1}{{\overset{\_}{v}}^{2}}{\int{\frac{\mathbb{d}u}{\overset{\_}{u} - u}\left\lbrack {\nabla_{T}{{\hat{g}}_{f}\left( {\lambda,u,w} \right)}} \right\rbrack}}}}} \middle| {}_{w = {\overset{\_}{w} + {\kappa{({u - \overset{\_}{u}})}}}}{{- \frac{1}{4\quad\pi^{2}}}{\int_{0}^{2\quad\pi}{\frac{{{D(\lambda)}\left( {{\kappa\quad{R(\lambda)}} - {a_{z}^{\prime}(\lambda)}} \right)} - {{R^{\prime}(\lambda)}\left( {\overset{\_}{w} - {\kappa\quad\overset{\_}{u}}} \right)}}{{\overset{\_}{v}}^{2}}{\chi\left( {\lambda,\overset{\_}{u}} \right)}{\int{\mathbb{d}{u\left( {\frac{\partial}{\partial w}{{\overset{̑}{g}}_{f}\left( {\lambda,u,w} \right)}} \right)}}}}}} \right|_{w = {\overset{\_}{w} + {\kappa{({u - \overset{\_}{u}})}}}}},{{{where}\nabla_{T}} = {{{D(\lambda)}\left( {{{R(\lambda)}\frac{\partial}{\partial u}} + {{a_{z}^{\prime}(\lambda)}\frac{\partial}{\partial w}}} \right)} + {{R^{\prime}(\lambda)}{\left( {{u\quad\frac{\partial}{\partial u}} + {w\quad\frac{\partial}{\partial w}} + 1} \right).}}}}$9. Special Example 2 and the detailed reconstruction steps9.1 Definition

We consider well-designed general saddles established by C-arm geometry(or ring geometry). The well-designed general saddle established by theC-arm geometry can be parameterized as (R cos α cos λ, R cos α sin λ, Rsin α)^(T), where α is a function of λ(α=α(λ)) and satisfies${\alpha } < {\frac{\pi}{2}.}$9.2 Detector Geometry

The coordinates for the detector are given in the reconstruction spaceas follows: $\left\{ \begin{matrix}{{{\overset{->}{e}}_{uc}(\lambda)} = \left( {{{- \sin}\quad\lambda},{\cos\quad\lambda},0} \right)^{T}} \\{{{\overset{->}{e}}_{vc}(\lambda)} = \left( {{{- \cos}\quad{\alpha cos}\quad\lambda},{{- \cos}\quad\alpha\quad\sin\quad\lambda},{{- \sin}\quad\alpha}} \right)^{T}} \\{{{\overset{->}{e}}_{wc}(\lambda)} = \left( {{{- \sin}\quad{\alpha cos}\quad\lambda},{{- \sin}\quad\alpha\quad\sin\quad\lambda},{\cos\quad\alpha}} \right)^{T}}\end{matrix} \right.$

The detector plane is arranged to be orthogonal to the vector {rightarrow over (e)}_(vc)(λ). A projection point of the point {right arrowover (α)}(λ) onto the detector plane is the origin of the detectorplane. Cartesian coordinates (u_(c),w_(c)) specify a pixel in thedetector plane. The u_(c)-axis is parallel to the vector {right arrowover (e)}_(u), and the w_(c)-axis is parallel to the vector {right arrowover (e)}_(w).

The distance between the detector plane and the X-ray source in theC-arm apparatus is denoted by D(>0), and the projection data on thedetector plane is denoted by g_(c)(λ,u_(c),w_(c)).

9.3 Reconstruction theorem

The reconstruction theorem is performed by the following steps.

STEP 1: Filtering

FF1: Differentiation

Computeg ₁(λ,u _(c) ,w _(c))=∇_(c) g _(c)(λ,u _(c) ,w _(c))where ∇_(c) is defined by$\nabla_{c}{= {\frac{\partial}{\partial\lambda} + {\frac{{\left( {D^{2} + u_{c}^{2}} \right)\cos\quad\alpha} + {D\quad w_{c}\sin\quad\alpha} + {u_{c}w_{c}\alpha^{\prime}}}{D}\frac{\partial}{\partial u_{c}}} + {\frac{{\left( {D^{2} + w_{c}^{2}} \right)\alpha^{\prime}} - {D\quad u_{c}\sin\quad\alpha} + {u_{c}w_{c}\cos\quad\alpha}}{D}{\frac{\partial}{\partial w_{c}}.}}}}$FF2: WeightingCompute${g_{2}\left( {\lambda,u_{c},w_{c}} \right)} = {\frac{D(\lambda)}{\sqrt{{D^{2}(\lambda)} + u_{c}^{2} + w_{c}^{2}}}{{g_{1}\left( {\lambda,u_{c},w_{c}} \right)}.}}$FF3: Interpolation before filteringg ₃(λ,u _(c) ,z)=g ₂(λ,u _(c) ,w _(c)(u _(c) ,z)),where z denotes the parameter for the filtering plane π(z), andw_(c)(u_(c),z) is an equation of the intersection line between thefiltering plane π(z) and the detector plane.FF4: 1D Hilbert transformComputeg₄(λ, u_(c), z) = ∫_(−∞)^(∞)𝕕u_(c)^(′)h_(H)(u_(c) − u_(c)^(′))g₃(λ, u_(c)^(′), z),where h_(H) is the impulse response of the Hilbert transform.FF5: Transform after filteringComputeg _(c) ^(F)(λ,u _(c) ,w _(c))=g ₄(λu _(c) ,z(u _(c) ,w _(c)))where z(u_(c),w_(c)) denotes the parameter for the filtering plane whichpasses through the point (u_(c),w_(c)) on the detector plane.STEP 2: BackprojectionCompute${{f\left( \overset{->}{x} \right)} = {\frac{1}{4\quad\pi}{\int_{0}^{2\quad\pi}{{\mathbb{d}\lambda}\quad\frac{1}{{\overset{\_}{v}}_{c}\left( {\overset{->}{x},\lambda} \right)}{g_{c}^{F}\left( {\lambda,{{\overset{\_}{u}}_{c}\left( {\overset{->}{x},\lambda} \right)},{{\overset{\_}{w}}_{c}\left( {\overset{->}{x},\lambda} \right)}} \right)}}}}},$where {right arrow over (u)}_(c)({right arrow over (x)},λ),{right arrowover (w)}_(c)({right arrow over (x)},λ) denote the coordinates of theintersection point of the detector plane and the line connecting thepoints {right arrow over (x)} and {right arrow over (α)}(λ), and{right arrow over (v)}_(c)({right arrow over (x)},λ)=R−x cos α cos λ−ycos α sin λ−z sin α.

Note that by eliminating the procedure FF5 of STEP 1 in the abovereconstruction steps and incorporating the corresponding procedure inthe backprojection of STEP 2, resolution in the reconstructed image canbe increased.9.4 The differentiation operation with respect to the parameter λ forthe well-designed general saddle curve can be eliminated from thereconstruction steps in 9.3 in the same way as 8.4. That is, we define{right arrow over (g)}_(c) by${{\overset{̑}{g}}_{c}\left( {\lambda,u_{c},w_{c}} \right)} = {\frac{D}{\sqrt{D^{2} + u_{c}^{2} + w_{c}^{2}}}{{g_{c}\left( {\lambda,u_{c},w_{c}} \right)}.}}$

Then, the reconstruction formula is given as follows:${{f\left( \overset{->}{x} \right)} = {\left. {{- \frac{1}{4\quad\pi^{2}}}{\int_{0}^{2\quad\pi}{{\mathbb{d}\lambda}\quad\frac{{DR}\quad\cos\quad\alpha}{{\overset{\_}{v}}_{c}^{2}}{\int{\frac{\mathbb{d}u_{c}}{\left( {{\overset{\_}{u}}_{c} - u} \right)^{2}}{{\hat{g}}_{c}\left( {\lambda,u_{c},w_{c}} \right)}}}}}} \middle| {}_{w_{c} = {{\overset{\_}{w}}_{c} + {\kappa_{c}{({u_{c} - {\overset{\_}{u}}_{c}})}}}}{{- \frac{1}{4\quad\pi^{2}}}{\int_{0}^{2\quad\pi}{{\mathbb{d}\lambda}\quad\frac{{DR}\left( {{\kappa_{c}\quad\cos\quad\alpha} - \alpha^{\prime}} \right)}{{\overset{\_}{v}}_{c}^{2}}{\int{\frac{\mathbb{d}u_{c}}{{\overset{\_}{u}}_{c} - u_{c}}\left( {\frac{\partial}{\partial w_{c}}{{\hat{g}}_{c}\left( {\lambda,u_{c},w_{c}} \right)}} \right)}}}}} \middle| {}_{w_{c} = {{\overset{\_}{w}}_{c} + {\kappa_{c}{({u_{c} - {\overset{\_}{u}}_{c}})}}}}{{- \frac{1}{4\quad\pi^{2}}}{\int_{0}^{2\quad\pi}{\frac{{DR}\left( {{\kappa_{c}\quad\cos\quad\alpha} - \alpha^{\prime}} \right)}{{\overset{\_}{v}}_{c}^{2}}{\chi_{c}\left( {\lambda,{\overset{\_}{u}}_{c}} \right)}{\int{\mathbb{d}{u_{c}\left( {\frac{\partial}{\partial w_{c}}{{\hat{g}}_{c}\left( {\lambda,u_{c},w_{c}} \right)}} \right)}}}}}} \middle| {}_{w_{c} = {{\overset{\_}{w}}_{c} + {\kappa_{c}{({u_{c} - {\overset{\_}{u}}_{c}})}}}}{{where}\quad{{\overset{\_}{v}}_{c}\left( {\overset{->}{x},\lambda} \right)}} \right. = {R - {x\quad\cos\quad\alpha\quad\cos\quad\lambda} - {y\quad\cos\quad\alpha\quad\sin\quad\lambda} - {z\quad\sin\quad\alpha}}}},{{{\overset{\_}{u}}_{c}\left( {\overset{->}{x},\lambda} \right)} = {\frac{D}{{\overset{\_}{v}}_{c}\left( {\overset{->}{x},\lambda} \right)}\left( {{{- x}\quad\sin\quad\lambda} + {y\quad\cos\quad\lambda}} \right)}},{{w_{c}\left( {\overset{->}{x},\lambda} \right)} = {\frac{D}{{\overset{\_}{v}}_{c}\left( {\overset{->}{x},\lambda} \right)}\left( {{{- x}\quad\sin\quad\alpha\quad\cos\quad\lambda} - {y\quad\sin\quad\alpha\quad\sin\quad\lambda} + {z\quad\cos\quad\alpha}} \right)}},{\kappa_{c} = \left\{ {{\begin{matrix}\frac{\left( {z - {R\quad\sin\quad\alpha}} \right)\cos\quad\lambda}{\sin\quad{\lambda\left( {{z\quad\sin\quad\alpha} - R} \right)}} & {{{if}\quad z} \geq {R\quad\sin\quad\alpha}} \\\frac{{- \left( {z - {R\quad\sin\quad\alpha}} \right)}\sin\quad\lambda}{\cos\quad{\lambda\left( {{z\quad\sin\quad\alpha} - R} \right)}} & {{{{if}\quad z} < {R\quad\sin\quad\alpha}},}\end{matrix}{\chi\left( {\lambda,\overset{\_}{u}} \right)}} = \left\{ \begin{matrix}\frac{\cos\quad{\alpha cos}\quad\lambda}{{D\quad\sin\quad\lambda} - {{\overset{\_}{u}}_{c}\cos\quad\alpha\quad\cos\quad\lambda}} & {{{if}\quad{\overset{\_}{w}}_{0}} > {D\quad\tan\quad\alpha}} \\\frac{{- \cos}\quad{\alpha sin}\quad\lambda}{{D\quad\cos\quad\lambda} - {{\overset{\_}{u}}_{c}\cos\quad{\alpha sin}\quad\lambda}} & {{{if}\quad{\overset{\_}{w}}_{0}} < {D\quad\tan\quad{\alpha.}}}\end{matrix} \right.} \right.}$9.5 Another example of formulae which eliminates the differentiationoperation with respect to the parameters of the well-designed generalsaddle curve from the reconstruction formula is given by${f\left( \overset{->}{x} \right)} = \left. {\frac{1}{4\quad\pi^{2}}{\int_{0}^{2\quad\pi}{{\mathbb{d}\lambda}\quad\frac{DR}{{\overset{\_}{v}}_{c}^{2}}{\int{\frac{\mathbb{d}u_{c}}{{\overset{\_}{u}}_{c} - u_{c}}\left\lbrack {\begin{pmatrix}{{\cos\quad\alpha\quad\frac{\partial}{\partial u_{c}}} +} \\{\alpha^{\prime}\frac{\partial}{\partial w_{c}}}\end{pmatrix}{{\hat{g}}_{c}\left( {\lambda,u_{c},w_{c}} \right)}} \right\rbrack}}}}} \middle| {}_{w_{c} = {{\quad\overset{\quad\_}{w}}_{c} + {\kappa_{c}{({u_{c} - {\overset{\quad\_}{u}}_{c}})}}}}{{- \frac{1}{4\quad\pi^{2}}}{\int_{0}^{2\quad\pi}{\frac{{DR}\left( {{\kappa_{c}\cos\quad\alpha} - \alpha^{\prime}} \right)}{{\overset{\_}{v}}_{c}^{2}}{\chi_{c}\left( {\lambda,{\overset{\_}{u}}_{c}} \right)}{\int{\mathbb{d}{u_{c}\left( {\frac{\partial}{\partial w_{c}}{{\hat{g}}_{c}\left( {\lambda,u_{c},w_{c}} \right)}} \right)}}}}}} \right|_{w_{c} = {{\overset{\_}{w}}_{c} + {\kappa_{c}{({u_{c} - {\overset{\_}{u}}_{c}})}}}}$

In the following, examples according to the present invention areexplained.

EXAMPLE 1

A first example of the present invention for a standard saddle given byx=R cos λ,y=R sin λ,z=h cos2λ.

The standard saddle is established by rotating an object (sample)clockwise about a rotation axis and moving the object up and down inaccordance with the equation given by z_(s)=−h cos2λ, where z_(s) is thez coordinate of a rotation stage on which the object is placed. In thiscase, the X-ray source and the detector are fixed (stationary). Asimilar result follows by rotating the object anticlockwise. FIG. 5shows an implementation example of a general saddle.

EXAMPLE 2

According to the present invention, a second example for the standardsaddle is given. Although the first example is easy to implement, theusage of the detector is not good. The usage of the detector isincreased by moving the detector up and down in accordance with theequation given by ${z_{D} = {\frac{D}{R}z_{S}}},$where z_(D) is the z coordinate of the detector.

EXAMPLE 3

According to the present invention, a third example is for the C-armsystem (or ring system).

There are two rotation axes in the C-arm geometry. A well-designedgeneral saddle is easily given by restricting the value (angle) of oneof the two rotation axes in the geometry by the value (angle) of theother axis.

C-arm geometry: There are two axes, λ-axis and α-axis in the C-armsystem, and the apparatus has two parameters λ and α. A circletrajectory is given by fixing the angular position of the α axis androtating the C-arm about the λ-axis. If the C-arm is rotated about thetwo axes, the coordinates of the X-ray source are given by (R cos αcosλ,R cos αsin λ,R sin α)^(T).

The trajectory becomes a well-designed general saddle, if the rotationabout the α-axis (α=α(λ)) satisfies certain conditions. The trajectoryof the X-ray source is a well-designed general saddle, if the rotationis restricted, for example, by${\alpha = {{\alpha(\lambda)} = {k\quad\cos\quad 2\quad\lambda}}},{0 < k < \frac{\pi}{2}},{or}$${\alpha = {{\alpha(\lambda)} = {\tan^{- 1}\frac{k\quad\cos\quad 2\quad\lambda}{\sqrt{1 - k^{2}}}}}},{0 < k < 1.}$

Refer to FIG. 7 showing the C-arm geometry.

The well-defined general saddle can be realized also in the ringgeometry instead of the C-arm geometry. The relative positionalrelationship between the X-ray source and the detector (detector plane)on the saddle trajectory is the same.

EXAMPLE 4

According to the present invention, a fourth example is given. Thegeneral saddle CT apparatus can reconstruct a thin (planar) object moreaccurately than oblique CT by avoiding directions along which the X-rayhardly penetrates the object, although the magnification cannot beincreased so much. The reason can be explained by the following threeaspects:

(1) The general saddle trajectory is capable of penetrating the objectin most of the directions as the oblique CT can.

(2) In the case of the general saddle, projection data obtained at amost slanted (lowest) angle can be utilized.

(3) The projection data of those projections which do not penetrate theobject can be neglected and need not be used in reconstruction. Althoughsome data lack in such a case, their effect is less severe for thegeneral saddle CT and more accurate images can be reconstructed than theoblique CT. Refer to FIG. 8 showing the geometry for reconstructing athin (planar) object with a general saddle.

EXAMPLE 5

According to the present invention, a fifth example is given. FIG. 9shows a block diagram of an apparatus for performing a method accordingto the present invention. A general saddle CT scan apparatus comprises ageneral saddle trajectory unit (for example, a dividing control unit fora C-arm system or ring system), a detector with a detector plane, anoperation control unit which receives detected data from these units andcontrols them, and a I/O and display unit. The operation control unitfurther comprises an X-ray source-detector system control unit, a CPU, amemory unit (which may include ROM, RAM, register and hard disk etc.), adifferentiating unit, a filtering unit, backprojection unit, and a threedimensional (3D) reconstruction unit, all of which are connected to abus. The operation unit comprises also an input/output interface andother commodity parts, which are not shown in the figure.

The differentiating, filtering, backprojection, and 3D reconstructionunits may be implemented by both software and/or hardware. Although notshown in the figure, each detailed step of the embodiment can beperformed one by one correspondingly with these units. Each step of themethod or operation steps according to the present invention memorizesprojection data (detected data) upon needs and transfer the data to thenext step.

Equations used for each step is memorized in a ROM beforehand, ormemorized, if necessary, in a hard disk or other high-speed memory andread out of it. A clock signal (not shown in the figure) controls thetotal system. First of all, a general saddle is specified and setthrough the I/O and display units.

After an object to be measured (sample) is placed at a predeterminedposition of the apparatus (for example, the center of a C-arm system),the initial position of the X-ray source is placed at a point a (λ) onthe predetermined general saddle trajectory. After a detector plane ofthe detector is placed to satisfy predetermined conditions, an X-raycone beam is irradiated onto the sample under control. The projection(image) which penetrates the object is detected on the detector plane ofthe detector and the detected data is memorized in accordance with thecoordinate system of the detector or the detector plane. The datadetected by the coordinate system of the detector plane is filteredalong an intersection line of a predetermined filtering plane and thedetector plane. A predetermined process of each step is performed on thedata by each processing unit before and after the filtering process. Thedata generated in each step is memorized in the memory and forwarded toa subsequent step.

The data after backprojection is memorized and stored as a basic data toreconstruct 3D images. The data is reconstructed to form a 3D imagethrough the 3D reconstruction unit and, if necessary, displayed on thedisplay using a visualization program. These operations are formed byCPU. The control signals are sent from CPU through I/O circuits (notshown in the figure), and the detection signals are output from thedetector. The positions of the X-ray source-detector system are alsodetected and utilized (fed back) to control their relative position.

According to the principle of present invention, there is provided a CTmethod which makes it possible to process CT slice images one by one.The CT method can be most easily implemented and reduces thecomputational time significantly. Each processing (slice) image ismemorized in the memory device in association with the parameter λ whichspecifies the location. Then the process proceeds to the next scanningstep (the next reconstruction step). In the same way, the steps arerepeated along the general saddle trajectory. When the X-ray sourcereturns to the initial position, one cycle is completed.

Since one cycle of the scan ends in a short time, X-ray exposure of anobject during one CT scan is reduced. There is also provided a fourdimensional CT with time as a parameter in principle, because one cycleof the CT scan ends in a short time and the X-ray source returns to theinitial position on the general saddle trajectory after one cycle.Therefore, a dynamic four dimensional CT which operates online in thetime sequence is made possible by the present invention, which bringsrevolution in the CT scan for moving objects such as heart (walls).

EXAMPLE 6

(Simulation Example 1)

According to the present invention, there is provided a sixth examplefor simulation use. The simulation studies have been done for theShepp-Logan phantom with the view differencing (VD) method and the viewindependent (VI) method based on general saddles according to (8.3) and(8.4) of the present invention, respectively. FIG. 10 shows crosssections of the reconstructed 3D images by each method at x=0, y=−0.768cm, and z=0.

In FIG. 10, VD1440 denotes the reconstructed images by the VD algorithmaccording to (8.3) of the present invention taken from projections inthe (1440) direction. VI 360 denotes the reconstructed images by the VIalgorithm according to (8.4) of the present invention taken fromprojections in the (360) direction. In the same way, VD 360 denotes thereconstructed images by the VD algorithm according to (8.3) of thepresent invention taken from projections in the (360) direction.

EXAMPLE 7

(Simulation Example 2)

According to the present invention, there is provided a seventh examplefor simulation use. The simulation studies have been done for a discphantom with the view differencing (VD) method and the view independent(VI) method based on a general saddle trajectory according to the (8.3)and (8.4) of the present invention, respectively. FIG. 1 shows crosssections (slices) of the reconstructed 3D images by each method at x=0,y=−0.768 cm, and z=0. In FIG. 11, VD1440 denotes the reconstructedimages by the VD algorithm according to (8.3) of the present inventiontaken from projections in the direction of (1440). VI 360 denotes thereconstructed images by the VI algorithm according to (8.4) of thepresent invention taken from projections in the (360) direction. In thesame way, VD 360 denotes the reconstructed images by the VD algorithmaccording to (8.3) of the present invention taken from projections inthe (360) direction.

EXAMPLE 8

(Simulation Example 3)

According to the present invention, there is provided an eighth examplefor simulation use. The simulation studies have been done for theShepp-Logan phantom with the view independent (VI) method based on awell-designed general saddle trajectory established by a C-arm geometrysystem according to (9.4) of the present invention. FIG. 12 shows crosssections of the reconstructed 3D images by each method at x=0, y=−0.768cm, and z=0. In FIG. 12, VI 720 denotes the reconstructed images by theVI algorithm according to (9.4) of the present invention taken fromprojections in the (720) direction.

The meritorious effects of the present invention are summarized asfollows.

According to the present invention, a general saddle scan method isproposed for CT imaging of a short object as well as partial CT imagingof a long object. Since the general saddle satisfies the cone-beam CTdata completeness condition, the general saddle CT can reconstruct theobject more accurately than the conventional cone-beam CT of thecircular trajectory. Moreover, since the general saddle scan avoids theproblem of overscan, it requires shorter measurement time and less X-rayexposure of objects than the helical scan. The general saddle scanmethod according to the present invention also makes it possible to scanan object serially along with time, because the relative positionbetween the X-ray source and the object returns to the original positionafter one rotation (cycle). Since the general saddle proposed in thepresent invention has wider applicability than the conventional saddles,it can be used in many easily realizable CT scan methods.

It should be noted that there had been no simple and effectivereconstruction algorithm even for the simplest saddle trajectory beforethe present invention. The present invention provides a filteredbackprojection (FBP)-type reconstruction algorithm for general saddlesfor the first time. Since the proposed algorithm is of the FBP-type, itis easy to implement and the computational speed is also accelerated.

The step of identifying a set of π-lines (according to the π-line basedmethods: Pack et al 2004, 2005, Xia et al 2005, Yu et al 2005, Zou et al2005) is not necessary to implement the proposed algorithms in thepresent invention. The exact FBP reconstruction algorithms are proposedfor the general saddle. The key idea of the proposed algorithm is thatthe filtering plane is parallel to one of the two horizontal axes whichdefine the general saddle. The proposed algorithms are theoreticallyexact, have a shift-invariant FBP structure, and do not depend from theconcept of π-line.

[Applicable Industrial Fields]

The present invention triggers the use of the general saddle as a scanmethod for industrial inspection and medical diagnostics. Since a pointon the general saddle returns to the original position after onerotation (cycle), serial scan of an object is also possible. Therefore,the general saddle is applicable to the following fields for examples.

1. Four dimensional cardiac CT (e.g., medical use)

2. Automatic inspection apparatus (e.g., industrial use)

The disclosures of articles of H. Yang, M. Li, K. Koizumi and H. Kudoentitled “Exact cone beam reconstruction for a saddle trajectory” (Phys.Med. Biol. 51 (2006) 1157-1172), and “View-independent reconstructionalgorithms for cone beam CT with general saddle trajectory”, Phys. Med.Biol. 51 (2006) 3865-3884 are incorporated herein by reference theretofor further detail and developments of the principles of the presentinvention in the theoretical aspects. The articles are coauthored by thepresent inventors and both published after the priority date of thepresent invention.

It should be noted that other objects, features and aspects of thepresent invention will become apparent in the entire disclosure and thatmodifications may be done without departing the gist and scope of thepresent invention as disclosed herein and claimed as appended herewith.

Also it should be noted that any combination of the disclosed and/orclaimed elements, matters and/or items may fall under the modificationsaforementioned.

1. A general saddle CT scan method comprising: (a) placing an object inassociation with a general saddle curve defined by the followingconditions; and (b) moving an X-ray source relatively to the objectalong the general saddle curve and detecting X-ray projection data ofthe object projected on a detector plane; wherein said saddle curve is asimple smooth curve which is single-connected, bounded and closedsatisfies the following conditions against a family of continuousparallel planes in a three dimensional space: (1) for any plane in thefamily of parallel planes, there exist four different intersectionpoints between the plane and the closed curve; (2) said fourintersection points form a rectangle; and (3) rectangles obtained forany two different planes in the family of parallel planes are parallelto each other.
 2. A general saddle CT scan and reconstruction methodcomprising: (a) placing an object in association with a general saddlecurve defined by the following conditions; (b) moving an X-ray sourcerelatively to the object along the general saddle curve and detectingX-ray projection data of the object projected on a detector plane; (c)filtering the projection data along a predetermined filtering planewhich is parallel to one of two major axes of the general saddle curve;and (d) performing three-dimensional reconstruction of the objectthrough backprojection, wherein said general saddle curve is a simplesmooth curve which single-connected, bounded and closed satisfies thefollowing conditions against a family of continuous parallel planes in athree dimensional space: (1) for any plane in the family of parallelplanes, there exist four different intersection points between the planeand the closed curve; (2) said four intersection points form arectangle; and (3) rectangles obtained for any two different planes inthe family of parallel planes are parallel to each other.
 3. The generalsaddle CT scan and reconstruction method according to claim 2, whereinthe filtering is performed along an intersection line between saidfiltering plane and the detector plane.
 4. The general saddle CT scanand reconstruction method according to claim 2, wherein the filteringplane Π(λ,{right arrow over (θ)}) is chosen to satisfy the followingconditions in relation to a point {right arrow over (α)}(λ) of the X-raysource on the general saddle curve: (1) the filtering plane passesthrough the point {right arrow over (α)}(λ); (2) the filtering plane isparallel to a vector {right arrow over (θ)}=(θ_(x),θ_(y),θ_(z))^(T)extending from the point {right arrow over (α)}(λ) to a point on thedetector plane; and (3) the filtering plane is parallel to a vector{right arrow over (e)}(λ,{right arrow over (θ)}), where the vector{right arrow over (e)}(λ,{right arrow over (θ)}) is chosen to satisfythe following condition:${\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} = \left\{ \begin{matrix}{{{\overset{->}{e}}_{x}\quad{if}\quad\theta_{z}} \geq 0} \\{{{\overset{->}{e}}_{y}\quad{if}\quad\theta_{z}} < 0.}\end{matrix} \right.$
 5. The general saddle CT scan and reconstructionmethod according to claim 2, wherein the filtering is performed by${{g_{F}\left( {\lambda,\overset{->}{\theta}} \right)} = \left. {\int_{- \pi}^{\pi}{{\mathbb{d}\gamma}\quad\frac{1}{\sin\quad\gamma}\frac{\partial}{\partial\overset{\_}{\lambda}}{p\left( {\overset{\_}{\lambda},{\xi\left( {\lambda,\theta,\gamma} \right)}} \right)}}} \right|_{\overset{\_}{\lambda} = \lambda}},{\theta \in S^{2}}$where${{\overset{->}{\xi}\left( {\lambda,\overset{->}{\theta},\gamma} \right)} = {{\cos\quad\gamma\quad\overset{->}{\theta}} + {\sin\quad\gamma\quad\frac{{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} - {\left( {{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} \cdot \overset{->}{\theta}} \right)\overset{->}{\theta}}}{{{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} - {\left( {{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} \cdot \overset{->}{\theta}} \right)\overset{->}{\theta}}}}}}},$and g(λ,{right arrow over (ξ)}) denotes the projection data.
 6. Thegeneral saddle CT scan and reconstruction method according to claim 2,wherein the backprojection is performed by${{f\left( \overset{->}{x} \right)} = {{- \frac{1}{4\quad\pi^{2}}}{\int_{0}^{2\quad\pi}{\frac{\mathbb{d}\lambda}{{\overset{->}{x} - {\overset{->}{a}(\lambda)}}}{g_{F}\left( {\lambda,\frac{\overset{->}{x} - {\overset{->}{a}(\lambda)}}{{\overset{->}{x} - {\overset{->}{a}(\lambda)}}}} \right)}}}}},$where f({right arrow over (x)}) denotes CT value at each point of theobject.
 7. The general saddle CT scan method according to claim 1,wherein a well-designed general saddle curve defined in the following isemployed as the general saddle curve in case where the object is long,and the reconstruction region FOV is sandwiched between two (horizontal)planes crossing the object, the general saddle curve being termed“well-designed” subject to the following conditions: (1) left (or right)part of a given general saddle curve above a first (horizontal) plane islocated on the left (or right) side of the reconstruction region FOV;and (2) front (or back) part of a given general saddle curve below asecond (horizontal) plane is located on the front (or back) side of afront (or back) plane of the reconstruction region FOV.
 8. The generalsaddle CT scan and reconstruction method according to claim 4, wherein athree-dimensional reconstruction of the object is performed by repeatingthe steps (c) and (d) for each value of the parameter λ of the generalsaddle curve.
 9. The general saddle CT scan and reconstruction methodaccording to claim 4, wherein the filtering comprises: differentiationof the projection data with respect to the parameter λ of the generalsaddle curve; weighting after the differentiation; interpolation beforefiltering; and 1D Hilbert transform of the interpolated data.
 10. Thegeneral saddle CT scan and reconstruction method according to claim 9,wherein filtering and coordinate transformation of the filtered data areperformed after the 1D Hilbert transform.
 11. The general saddle CT scanand reconstruction method according to claim 9, wherein thebackprojection is performed using filtering and coordinatetransformation of the filtered data after the 1D Hilbert transform. 12.The general saddle CT scan and reconstruction method according to claim4, wherein the filtering is performed by defining two weighted databased on the projection data, and the reconstruction is performed usingthese data without differentiation operation with respect to theparameter λ of the general saddle curve.
 13. The general saddle CT scanmethod according to claim 1, wherein a well-designed general saddlecurve is produced by restricting value about one axis among two rotationaxes in C-arm geometry or ring geometry with respect to a value of theother axis.
 14. The general saddle CT scan method according to claim 1,wherein the general saddle curve is established by rotating the objectabout a rotation axis thereof, fixing either one of the X-ray source andthe detector plane, and controlling their relative positionalrelationship.
 15. The general saddle CT scan method according to claim1, wherein the general saddle curve is established by fixing the X-raysource and the detector plane, rotating the object about its rotationaxis and moving the object forward and reverseward along a direction ofits rotation axis in accordance with a predetermined equation.
 16. Acomputer program product for executing the method of claim 1 by acomputer.
 17. A general saddle CT scan apparatus comprising: (a) asupport unit which supports an object relatively to a general saddlecurve defined by the conditions below; (b) a driving unit for an X-raysource-object-detector system, which provides relative movement of anX-ray source and a detector plane along a general saddle curverelatively to the object; and (c) a capture unit which detectsprojection data of X-ray radiated from the X-ray source and penetratedthrough the object; wherein said general saddle curve is defined as suchif a simple smooth curve which is single-connected, bounded and closedsatisfies the following conditions against a family of continuousparallel planes in a three dimensional space: (1) for any plane in thefamily of parallel planes, there exist four different intersectionpoints between the plane and the closed curve; (2) said fourintersection points form a rectangle; and (3) rectangles obtained forany two different planes in the family of parallel planes are parallelto each other.
 18. A general saddle CT scan apparatus comprising: (a) asupport unit which supports an object relatively to a general saddlecurve defined by the conditions below; (b) a driving unit for an X-raysource-object-detector system, which provides relative movement of anX-ray source and a detector plane along a general saddle curverelatively to the object; (c) a capture unit which detects projectiondata of X-ray radiated from the X-ray source and penetrated through theobject; (d) a filtering unit that filters the projection data along apredetermined filtering plane which is parallel to one of two major axesof the general saddle curve; and (e) a back projection andreconstruction unit that performs reconstruction of the object throughbackprojection; wherein said general saddle curve is defined as such ifa simple smooth curve which is single-connected, bounded and closedsatisfies the following conditions against a family of continuousparallel planes in a three dimensional space: (1) for any plane in thefamily of the parallel planes, there exist four different intersectionpoints between the plane and the closed curve; (2) said fourintersection points form a rectangle; and (3) rectangles obtained forany two different planes in the family of parallel planes are parallelto each other.
 19. The general saddle CT scan apparatus according toclaim 17, wherein the filtering unit performs the filtering along anintersection line of a filtering plane and a detector plane.
 20. Thegeneral saddle CT scan apparatus according to claim 17, wherein afiltering plane Π(λ,{right arrow over (θ)}) is chosen to satisfy thefollowing conditions in relation a point of the X-ray source {rightarrow over (α)}(λ) on the general saddle curve: (1) the filtering planepasses through the point {right arrow over (α)}(λ); (2) the filteringplane is parallel to a vector {right arrow over(θ)}=(θ_(x),θ_(y),θ_(z))^(T) extending from the point {right arrow over(α)}(λ) to a point on the detector plane; and (3) the filtering plane isparallel to a vector {right arrow over (e)}(λ,{right arrow over (θ)}),where the vector {right arrow over (e)}(λ,{right arrow over (θ)}) ischosen to satisfy the following condition:${\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} = \left\{ \begin{matrix}{{{\overset{->}{e}}_{x}\quad{if}\quad\theta_{z}} \geq 0} \\{{{\overset{->}{e}}_{y}\quad{if}\quad\theta_{z}} < 0.}\end{matrix} \right.$
 21. The general saddle CT scan apparatus accordingto claim 18, wherein the filtering is performed by${{g_{F}\left( {\lambda,\overset{->}{\theta}} \right)} = \left. {\int_{- \pi}^{\pi}{{\mathbb{d}\gamma}\quad\frac{1}{\sin\quad\gamma}\frac{\partial}{\partial\overset{\_}{\lambda}}{p\left( {\overset{\_}{\lambda},{\xi\left( {\lambda,\theta,\gamma} \right)}} \right)}}} \right|_{\overset{\_}{\lambda} = \lambda}},{\theta \in S^{2}}$where${{\overset{->}{\xi}\left( {\lambda,\overset{->}{\theta},\gamma} \right)} = {{\cos\quad\gamma\quad\overset{->}{\theta}} + {\sin\quad\gamma\quad\frac{{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} - {\left( {{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} \cdot \overset{->}{\theta}} \right)\overset{->}{\theta}}}{{{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} - {\left( {{\overset{->}{e}\left( {\lambda,\overset{->}{\theta}} \right)} \cdot \overset{->}{\theta}} \right)\overset{->}{\theta}}}}}}},$and g(λ,{right arrow over (ξ)}) denotes the projection data.
 22. Thegeneral saddle CT scan apparatus according to claim 18, wherein thebackprojection is performed by${{f\left( \overset{->}{x} \right)} = {{- \frac{1}{4\quad\pi^{2}}}{\int_{0}^{2\quad\pi}{\frac{\mathbb{d}\lambda}{{\overset{->}{x} - {\overset{->}{a}(\lambda)}}}{g_{F}\left( {\lambda,\frac{\overset{->}{x} - {\overset{->}{a}(\lambda)}}{{\overset{->}{x} - {\overset{->}{a}(\lambda)}}}} \right)}}}}},$where f({right arrow over (x)}) denotes CT value at each point of theobject.
 23. The general saddle CT scan apparatus according to claim 18,wherein a well-designed general saddle curve defined in the following isemployed as the general saddle curve, in case where the object is long,and the reconstruction region FOV is sandwiched between two (horizontal)planes crossing the object, the general saddle curve being termed“well-designed” subject to the following conditions: (1) left (or right)part of a given general saddle curve above a first (horizontal) plane islocated on the left (or right side) of the reconstruction region FOV;and (2) front (or back) part of a given general saddle curve below asecond (horizontal) plane is located on the front (back) side of a front(or back) plane of the reconstruction region FOV.
 24. The general saddleCT scan apparatus according to claim 18, further comprising a memoryunit that stores the backprojection and reconstruction data from theback projection and reconstruction unit (e); and a three-dimensionalreconstruction unit that reconstructs the object from the packprojection and reconstruction data.
 25. The general saddle CT scanapparatus according to claim 17, wherein the well-designed generalsaddle curve is produced by restricting value about one axis among tworotation axes in C-arm geometry or ring geometry with respect to a valueof another axis.
 26. The general saddle CT scan apparatus according toclaim 17, wherein the general saddle curve is established by rotatingthe sample about a rotation axis thereof, fixing either one of the X-raysource and the detector plane, and controlling their relative positionalrelationship.
 27. The general saddle CT scan apparatus according toclaim 17, wherein the general saddle curve is established by fixing theX-ray source and the detector plane, rotating the sample about arotation axis and moving the object forward and reverseward along adirection of its rotation axis in accordance with a predeterminedequation.